Official Solution:If \(xy ≠ 0\), what is the value of \(\frac{x}{|y|}\) ? (1) \(|x| = |y|\)
The above means that \(x\) and \(y\) have the same magnitude but we don't know the sign of \(x\) to determine the value of \(\frac{x}{|y|}\). If \(x < 0\), then \(\frac{x}{|y|}=-1\) (for example, consider \(x=-1\) and \(y=1\)) but if \(x > 0\), then \(\frac{x}{|y|}=1\) (for example, consider \(x=1\) and \(y=1\)). Not sufficient.
(2) \(y = |x|\)
The above means that \(y > 0\) (it equals to the absolute value of non-negative number \(x\)) but again we don't know the sign of \(x\). If \(x < 0\), then \(\frac{x}{|y|}=-1\) (for example, consider \(x=-1\) and \(y=1\)) but if \(x > 0\), then \(\frac{x}{|y|}=1\) (for example, consider \(x=1\) and \(y=1\)). Not sufficient.
(1)+(2) We know that \(x\) and \(y\) have the same magnitude and that \(y > 0\). Still nothing about the sign of \(x\). Not sufficient.
Answer: E